The Rules of Logic Part 4: The Laws of Noncontradiction and Transitive Properties

The two most fundamental rules of logic are the Law of Noncontradiction and the Law of Transitive Properties. In fact, all of the other rules of logic stem from these two laws. Both laws are very simple and easy to understand, yet people frequently ignore or misuse them. Therefore I will attempt to explain how they actually work.

Law of Noncontradiction

This law simply states that something cannot be A and not A simultaneously. In other words, two mutually exclusive things cannot exist at the same time. So, for example, the Law of Noncontradiction tells us that it is impossible for both an immovable object and an unstoppable force to exist simultaneously, because each one cancels the other out. In other words, if there is an object that is truly immovable, then there cannot be an unstoppable force because this would contradict the properties of the immovable object. Conversely, if there is an unstoppable force, then there cannot be an immovable object because it would contradict the properties of the unstoppable force.

Beyond unraveling fun philosophical paradoxes, this law has some very real and useful applications. It is, in fact, the reason that math works consistently. Let me use an example from a previous post.

  1. The sum of the angles of any triangle = 180 degrees
  2. For triangle ABC, angle A = 90
  3. For triangle ABC, angle B= 45
  4. Therefore, for triangle ABC, angle C = 45

Why is the conclusion valid? Quite simply, its valid because the Law of Noncontradiction states that it is not possible for all of these premises to be true and for angle C to be anything other than 45 degrees. In other words, if C was anything other than 45, it would contradict the other premises, therefore C must be 45 degrees.

This brings me to the most important and practical application of this law. Because of the Law of Noncontradiction, you have to use consistent logic in your arguments. In other words, your arguments and beliefs cannot conflict with one another. For example, suppose that I said, “Indiana Jones and the Raiders of the Lost Ark is one of the best movies ever because Harrison Ford is in it” (i.e., I am asserting the Harrison Ford’s presence alone is the reason that it is one of the best movies ever). You then respond with, “he is also in Kingdom of the Crystal Skull, is it one of the best movies ever?” According to the Law of Noncontradiction, I have to either claim that Kingdom of the Crystal Skull is one of the best movies ever, or I have to reject my previous claim and acknowledge that Ford’s presence alone is not enough to qualify a movie as one of the greatest of all time. I cannot simultaneously claim that Raiders is great simply because Ford is in it and that Kingdom sucks even though Ford is in it. To be clear, its fine to say Ford’s acting was good in Raiders but bad in Kingdom, but my argument was that his presence alone was enough to make Raiders a great movie, and it is that argument which is clearly flawed.

Law of Transitive Properties

This law simply states that the properties of one premise must transition or carry over to the other premises. This is the law that you apply when trying to solve simple logical puzzles. For example:

Bob is taller than Bill. Bob is shorter than Tom. Jane is taller than Tom. Is Jane taller than Bill?

We can rearrange these facts into premises in a syllogism:

  1. Jane is taller than Tom
  2. Tom is taller than Bob
  3. Bob is taller than Bill
  4. Therefore, Jane is taller than Bill

You see, the property of height transitioned to each premise, resulting in the conclusion that Jane must be taller than Bill. You can also demonstrate this law using math examples.

  1.  2+2 = 4
  2.  4+4 = 8
  3.  8+8 = 16
  4.  Therefore, 2+2+4+8 = 16

This law can also be applied to equivalencies. For example:

  1. A=B
  2. B=C
  3. Therefore, A = C

It is important to note that this law only works with true premises. In other words, if one of the properties is incorrect, then the argument will not work, and it will often result in a slippery slope fallacy. For example:

  1. If you use marijuana you will start doing other drugs
  2. If you start doing other drugs you will die of an overdose
  3. Therefore, using marijuana will make you die of an overdose

The problem here should be obvious, neither premise is consistently true, therefore the argument does not work. A closely related problem occurs when premises are true under certain conditions. For example:

  1. Under certain conditions A causes B
  2. Under a different set of conditions B causes C
  3. Under a different set of conditions C causes D
  4. Therefore A causes D

The problem here is that A will only lead to D if all three of the precise conditions are met. You could say that, “under a very precise serious of changing conditions A will cause D” but more often than not, the odds of the conditions changing in the right way for A to cause D are so low that it isn’t work worrying about A causing D. For example:

  1. GMOs create novel combinations of genes
  2. Under certain conditions, bacteria in your gut can incorporate DNA from your food into their genome
  3. In some cases, when bacteria take in new DNA they can form new proteins
  4. Under some conditions some proteins are toxic
  5. Therefore GMOs create toxic proteins

I intend on dealing with this argument in more detail in a later post, but for now my point is simply that the conclusion is not valid because the odds of this sequence actually playing out are extremely low (it is a slippery slope fallacy). Also, there is no reason why this couldn’t happen with non-GMO food, so this argument actually violates the Law of Noncontradiction as well.

Other posts on the rules of logic:



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